Shafieepoorfard and Raginsky, have studied the problem of controlling a Markov Decision Process while the agent is subject to observation constraint [1]}. In particular, the observations of the agent here are considered to have a limited mutual information with the state of the system. Most of the literature in control theory society either treats a given medium for observations as a given constraint or considers transmissions of the sensory system as an additional cost [2]}. The work in [1]} together with [4]} can be considered among the very first few papers which solve the problem of finding a stochastic control function in conjunction with the control problem. The problem of joint design of the observation and control policy is studied in this paper in its very general form as the policies are not considered as deterministic but as stochastic functions, where the transitions of the system are also considered to be stochastic. The problem of one-shot control-communication policy optimization under mutual information constraint, is first shown by authors to be a form of rate-distortion problem. Armed with this analytical result, authors consequently use the Bellman equations to reformulated the core problem as a one-shot control policy optimization under mutual information constraint. This particular way that the infinite horizon control problem under observation constraint is formulated by [1]} was first introduced by Sims in his seminal work [6]} which won him a Nobel prize.
